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Affine Gelfand-Dickey brackets and holomorphic vector bundles Pavel I. Etingof and Boris A. Khesin Yale University Department of Mathematics New Haven, CT 06520 USA e-mail: etingof @ math.yale.edu khesin @ math.yale.edu 4 9 9 hep-th 9312123 1 Abstract n a J We define the (second) Adler-Gelfand-Dickey Poisson structure on differential 7 operators over an elliptic curve and classify symplectic leaves of this structure. 2 This problem turns out to be equivalent to classification of coadjoint orbits for v double loop algebras, conjugacy classes in loop groups, and holomorphic vector 3 bundles over the elliptic curve. We show that symplectic leaves have a finite but 2 1 (unlike the traditional case of operators on the circle) arbitrarily large codimension, 2 and compute it explicitly. 1 3 Introduction 9 / In the seventies M.Adler[A], I.M.Gelfand and L.A.Dickey [GD] discovered a nat- h t ural Poisson structure on the space of n-th order differential operators on the circle - p with highest coefficient 1 which is now called the (second) Gelfand-Dickey bracket. e h This bracket arises in the theory of nonlinear integrable equations under various : v names (nKdV-structure, classical Wn-algebra). B.L.Feigin proposed to consider i and study symplectic leaves for the Gelfand-Dickey bracket – a problem motivated X by the fact that for n = 2 these symplectic leaves are orbits of coadjoint rep- r a resentation of the Virasoro algebra. A classification of symplectic leaves for the Gelfand-Dickey bracket and a description of their adjacency were given in [OK]. It turned out that locally symplectic leaves are labeled by one of the following: 1) conjugacy classes in the group GL ; n 2) orbits of the coadjoint representation of the affine Lie algebra gl ; n 3) equivalence classes of flat vector bundles on the circle of rank n (these three c things are in one-to-one correspondence). Moreover, adjacency of symplectic leaves is the same as that for conjugacy classes, orbits and vector bundles. Finally, the codimension of a symplectic leaf is equal to any of the following: 1) the dimension of the centralizer of the corresponding conjugacy class; 2) the codimension of the corresponding coadjoint orbit; 3) the dimension of the space of flat global sections of the bundle of endomor- phisms of the corresponding flat vector bundle. In Section 1 of this paper we define an “affine” analogue of the Gelfand-Dickey bracket. It is realized on the space of n-th order differential operators on an elliptic curve which are polynomials in ∂ with smooth coefficients and highest coefficient 1. The reason to consider such brackets is a search for an appropriate two-dimensional counterpart of the theory of affine Lie algebras. One can show that the “affine” analogue of the Drinfeld-Sokolov reduction [DS] sends the linear Poisson bracket 2 on the double loop algebra (cf.[EF]) into the quadratic Gelfand-Dickey bracket on the space of differential operators on the elliptic curve. The main goal of the paper is to classify and study the symplectic leaves of the affine Gelfand-Dickey bracket. In the case n = 2, the problem of classification of symplectic leaves coincides with the problem of classification of orbits of the coadjoint representation of the complex Virasoro algebra defined in [EF] – the Lie algebra of pairs (f,a)where f is a smooth function on an elliptic curve M and a is a 3 complex number, with the commutation law [(f,a)(g,b)]= f∂g −g∂f, f∂ g . M In Section 2 we show that locally symplectic leaves of this(cid:0)bracket are Rlabeled b(cid:1)y 1) Conjugacy classes for the action of the loop group LGL (C) on the semidirect n product of C∗⋉LGL (C) (where LGL (C) denotes the connected component of n 0 n 0 the identity in the group LGL (C)); n 2) orbits of the coadjoint representation of the “double”’ affine Lie algebra – a central extension of the Lie algebra of gl -valued smooth functions on the elliptic n curve[EF]; 3)equivalence classes ofholomorphic vectorbundles ofrank n anddegree zero on the elliptic curve (as before, these three things are in one-to-one correspondence). Since holomorphic vector bundles over an elliptic curve are completely classified [At], this result gives a good description of symplectic leaves. In Section 3 we show that, similarly to the classical case, adjacency of symplectic leaves in the affine case is the same as for conjugacy classes, orbits and vector bundles, and that the codimension of a symplectic leaf is equal to 1) the dimension of the centralizer of the corresponding conjugacy class; 2) the codimesion of the corresponding coadjoint orbit; 3) the dimension of the space of holomorphic sections of the bundle of endomor- phisms of the corresponding holomorphic vector bundle. In particular, this implies that in the affine case the codimension of a symplectic leaf, though always finite, can be arbitrarily large, unlike the finite dimensional case, in which it is bounded from above by dimGL = n2. n These results constitute a two dimensional (or affine) counterpart of the results of [OK] for Gelfand-Dickey brackets. Similarly to the non-affine case, they can be generalized to other classical Lie groups – SL , Sp , SO (see [OK]). n 2n 2n+1 The key toolin the study of Gelfand-Dickey brackets is the notionof monodromy of a differential operator. For the case of an elliptic curve the monodromy is a conjugacy class in the affine GL (more precisely, in the one-dimensional extension n C∗⋉LGL (C) of the loop group of GL ). This justifies the name “affine Gelfand- n 0 n Dickey bracket”. In Section 4 of the paper we discuss the question whether the map assigning an equivalence class of vector bundles to a symplectic leaf is surjective. This question is equivalent to the question whether any monodromy can be realized by an n-th order differential operator. For the usual Gelfand-Dickey bracket the answer to this question is positive (it follows, for example, from the results of M.Shapiro [S]). We prove that the answer is positive in the affine case as well, and describe an explicit realization for n = 2 using Atiyah’s classification of vector bundles over an elliptic curve. In Section 5 we discuss the global structure of the fibration of the space of differential operators by symplectic leaves, which in the classical case is defined geometrically by homotopy classification of quasiperiodic nonflattening curves on 3 a real projective space [O,OK,KS]. It turns out that the problem of counting symplectic leaves of the affine GL -Gelfand-Dickey bracket corresponding to the trivial 2 rank 2 vector bundle leads to a nice topological problem of classification of nowhere holomorphic maps from an elliptic curve to the complex projective line up to homotopy. In the GL case we encounter the problem of homotopy classification of n maps f from an elliptic curve to CPn−1 such that the vectors ∂f,...,∂n−1f are everywhere linearly independent. These maps are the affine counterparts of nonflattening curves in RPn−1. At the moment a complete solution of this problem (even in the GL -case) is unknown to the authors. 2 Acknowledgements The authors are grateful to V.Arnold, I.Frenkel, and R.Montgomery for useful remarks. 1. Gelfand-Dickey brackets. WestartbyrecallingthedefinitionoftheGelfand-Dickeystructures(see[A,GD,DS]). Let M be a compact smooth orientable closed manifold, k = R or C, C∞(M,k) be the algebra of smooth k-valued functions on M, ω be a volume form on M. Let D be a differential operator on C∞(M,k) such that (Df)ω = 0 and D(fg) = M (Df)g+f(Dg) for any f,g ∈ C∞(M,k). R Define the vector space L˜ as follows: n−1 (1.1) L˜= {P = u Dm|u ∈ C∞(M,k)}. m+1 m mX=0 To realize the dual space to L˜, we need to introduce pseudodifferential symbols. They areformal expressions oftheform ∞ a D−m, m ∈ Z, a ∈ C∞(M,k). m=m0 m 0 m It is known that such symbols form aPn associative algebra: two symbols A,B can be multiplied with the help of the rules D ◦ f = f ◦ D + Df, D−1 ◦ f = f ◦D−1 −f′ ◦D−2 +f′′ ◦D−3 −..., for any f ∈ C∞(M,k). We realize (the regular part of) the dual space to L˜ as follows: n (1.2) A = {A = a D−m|a ∈ C∞(M,k)}, m m mX=1 and the pairing L˜⊗A → k is given by the formula (1.3) < P,A >= Res(PA)ω, Z M where Res(X) is the coefficient to D−1 in a pseudodifferential operator X. It is clear that any regular linear functional on L˜ has this form. Note that Res(PA−AP) = Df, where f is some function on M, which implies that Res(PA)ω = Res(AP)ω. M M LeRt L be the affineR space of all operators of the form L = Dn + P, P ∈ L˜. Clearly, the tangent space to L at any point is naturally identified with L˜. 4 Following Adler, Gelfand and Dickey, let us assign a vector field V on L to A every regular linear functional A on L˜. Its value at a point L ∈ L is: (1.4) V (L) = L(AL) −(LA) L, A + + where X denotes the differential part of X. + Let C denote the algebra of smooth functions on L for k = R, and the algebra of holomorphic functions on L for k = C. Then assignment (1.4) allows one to define a Poisson bracket on C: (1.5) {f,g}(L) =< dg | ,V (L) > . L df|L Let us call this bracket the Gelfand-Dickey (GD) bracket. It equips L with a structure of a Poisson manifold. Let us now define symplectic leaves of the GD bracket and their codimensions. Let L ∈ L. A vector v ∈ T L = L˜ is called a Hamiltonian vector if there exists L A ∈ A such that v = V (L). A Define the symplectic leaf O to be the set of all points L′ ∈ L such that there L exists a smooth curve γ : [0,1] → L such that γ(0) = L, γ(1) = L′, and dγ is dt a Hamiltonian vector for any t. It is clear that two symplectic leaves are either disjoint or identical. Therefore, the space L becomes a disjoint union of symplectic leaves. The tangent space T O ⊂ L˜ to the symplectic leaf O at L is obviously the L L L space of all Hamiltonian vectors at L. Define the codimension of O to be the L codimension of this tangent space in L˜. This definition makes sense because the codimension of a symplectic leaf is the same at all its points. We will be concerned with the following two special cases of GD brackets. Main definition. Case 1. M = S1, k = R or C, D = d , ω = dx. The GD bracket corresponding dx to this situation is called the GL (k)-GD bracket [GD]. n Case 2. M is a nondegenerate elliptic curve over C: M = C/Γ, where Γ is a lattice generated by 1 and τ, where Im τ > 0, k = C, D = ∂ = ∂ = 1( ∂ +i ∂ ), ∂z¯ 2 ∂x ∂y where z = x+iy is the standard complex coordinate on C, ω = idz∧dz¯. The space 2 L consists of differential operators ∂n + n−1u (z,z¯)∂j, where u ∈ C∞(C/Γ). j=0 j+1 i We call the GD bracket corresponding toPthis case the affine GL -GD bracket. n Symplectic leaves of the GL -GD bracket are described in[OK]. In this paper n a similar description is given for symplectic leaves of the affine GL -GD bracket. n To emphasize the parallel between the non-affine and affine theories, we give an exposition of both of them, marking definitions and statements from the non-affine theory by the letter A and from the affine theory by the letter B. 5 2. Local classification of symplectic leaves Definition 1AB. Let f = (f ,...,f ) be a smooth kn-valued function on some 1 n covering of M (k = R or C). The matrix-valued function W(f) = (w ), w = ij ij Di−1f is called the Wronski matrix of f. j Proposition 1A. LetL be a differentialoperatorof the formL = dn + n−1u dj , dxn j=o j+1dxj u ∈ C∞(S1,k). Then: P j (i) there exists a set of n solutions f = (f ,...,f ) of the equation Lφ = 0 1 n belonging to C∞(R,k) whose Wronski matrix is everywhere nondegenerate (here R is regarded as a cover of S1); (ii) if ˜f = (f˜,...,f˜ ) is another set of solutions satisfying (i) then there exists a 1 n unique matrix R ∈ GL (k) such that ˜f = fR; n (iii) if f = (f ,...,f ) is any set of smooth k-valued functions on the real line such 1 n that its Wronski matrix is everywhere nondegenerate, and if f(x+1) = n f(x)R i=1 for some R ∈ GL (k), then there exists a unique differential operatorPL = dn + n dxn n−1u dj with periodic coefficients such that Lf = 0 for all i. j=0 j+1dxj i P Proof. This is a standard statement from the theory of ordinary differential equations. (cid:3) Let Σ = C/Z be a cylinder. This cylinder has a natural structure of an abelian group, is equivalent to C∗ as a complex manifold, and naturally covers the elliptic curve M = C/(Z⊕τZ). From now on we do not make a distinction between Σ and C∗. Before we formulate the affine analogue of Proposition 1A, we need to define loop groups. We will need three versions of a loop group for GL (C): n Notation. LGL (C) is the group of holomorphic GL (C)-valued functions on Σ. n n LGL (C) is the connected component of identity in LGL (C). GL (C) is the n 0 n n semidirect product Σ ⋉ LGL (C) , where Σ acts on LGL (C) by (z ◦ g)(w) = n 0 n 0 g(w+z). The group GL (C) should be regarded as the group of pairs (g(·),τ), g ∈ n LGL (C) , τ ∈ Σ, with the multiplicationlaw (g(z),τ)(h(z),θ)= (g(z)h(z+τ),τ+ n 0 θ). It is clear that LGL (C) is embedded into GL (C) by the map g(·) → (g(·),0). n 0 n ConsidertheactionofLGL (C) onGL (C)by conjugacy. Wewillcallthe orbits n n of this action restricted conjugacy classes. Proposition 1B. LetL be a differentialoperatorof the formL = ∂n+ n−1u ∂j, j=0 j+1 u ∈ C∞(M,C), where M is an elliptic curve. Then: P j (i) there exists a set of n solutions f = (f ,...,f ) of the equation Lφ = 0 1 n belonging to C∞(Σ,C) whose Wronski matrix is everywhere nondegenerate (here Σ is regarded as a cover of M); (ii) if ˜f = (f˜,...,f˜ ) is another set of solutions satisfying (i) then there exists a 1 n unique matrix R(z) ∈ LGL (C) such that ˜f = fR. n (iii) if f = (f ,...,f ) is any set of smooth complex-valued functions on Σ such 1 n that its Wronski matrix is everywhere nondegenerate, and if f(z + τ) = f(z)R(z) for some R(z) ∈ LGL (C), then there exists a unique differential operator L = n n n−1 j ∂ + u ∂ such that Lf = 0 for all i. j+1 i P 6 Proof. First of all, statements (i) and (ii) are true in a small enough neighborhood U of every point p ∈ Σ [AB]. Let gp = (gp,....,gp) be the corresponding sets of p 1 n p solutions. By thelocalversion ofstatement (ii), whenever U and U intersect, g = p q j n gqQpq, where Qpq(z) are holomorphic GL (C)-valued functions on U ∩U . i=1 i ij n p q TPhese functions satisfy the conditions: QpqQqp = 1, QpqQqrQrp = 1, which imply that they are clutching transformations of some holomorphic vector bundle E of L rank n on Σ. SinceΣisequivalenttoC∗ asacomplexmanifold, anyholomorphicvectorbundle over Σ has to be trivial. This, of course, applies to E , which implies that E has L L n global holomorphic sections s ,...,s which are everywhere linearly independent. 1 n That is to say, for every p ∈ Σ there exists a holomorphic function Sp(z) on U p with values in GL (C) such that Sp = QpqSq on U ∩U for any p,q ∈ Σ (s are the n p q i p p p p q columnsofS). Therefore, thefunctionsf = g S satisfytheconditionf = f j i i ij j j on U ∩U . This means, we have a globally dPefined vector-function f = (f ,...,f ), p q 1 n p p such that f | = f . Since the functions S (z) are holomorphic, the functions f j Up j ij j satisfy the equation Lf = 0. Also, W(f) = W(gp)Sp in every U , which implies j p W(f) is everywhere nondegenerate. This settles (i). Now let φ be any smooth complex function on Σ. Consider the column vector n−1 Φ = (φ,∂φ,...,∂ φ). It is obvious that φ is a solution of Lφ = 0 if and only if Φ satisfies the first order n × n-matrix equation ∂Φ = A Φ, where A is the L L Frobenius matrix corresponding to L: 0 1 ... ... 0 1 j −i = 1 0 0 1 ... 0   (2.1) A = ... ... ... ... ... , i.e. (A ) =  −u i = n L L ij j   0 0 ... ... 1     0 otherwise −u −u ... ... −u   1 2 n   This implies that if f = (f ,...,f ) is a set of solutions to Lφ = 0 then the Wronski 1 n matrix W(f) satisfies the equation (2.2) ∂W = A W. L To prove (ii), define the matrix function R on Σ by W(˜f) = W(f)R. This matrix is obviously always in GL (C), and it is holomorphic on Σ because both W(˜f) and n W(f) satisfy the equation ∂W = A W. Thus, R ∈ LGL (C). L n To establish (iii), for any f satisfying the conditions of (iii) define the vector- function u = (u ,...,u ) on Σ by the formula 1 n n (2.3) ∂ f +uW(f) = 0. Thisvectorfunction existsand isunique because ofthe nondegeneracy of W. More- n over, it is τ-periodic since both ∂ f and W(f) multiply by R from the right when n n−1 j z is replaced by z +τ. Now set L = ∂ + u ∂ . It is obvious that (2.3) is j=0 j+1 equivalent to the condition that Lf = 0 foPr all i, which implies (iii). (cid:3) i Propositions 1A and 1B have a simple geometric reformulation: Proposition 1AB. For every vector-function f with a nondegenerate Wronski matrix there exists a unique differential operator Lf ∈ L such that Lffi = 0, and the assignment f → Lf is a principal fibration over L whose fiber is GLn(k) in Case 1 and LGL (C) in Case 2. n 7 Corollary 2AB. Let L(t) be any smooth curve in L. Then there exists a smooth family of vector-functionsft with a nondegenerate Wronskimatrixsuch thatL(t)ft = i 0 for all i and for all t. Proof. This is just the statement that any path on the base of a fiber bundle can be covered by a path on the total space. (cid:3) Let us now define the notion of monodromy of a differential operator. Definition2A. Let L be a differentialoperatorof the formL = dn + n−1u dj , dxn j=0 j+1dxj u ∈ C∞(R/Z,k). Let f = (f ,...,f ) be a set of solutions of the equPation Lφ = 0 j 1 n belonging to C∞(R,k) whose Wronski matrix is everywhere nondegenerate. Let R ∈ GL (k) be the matrix such that f(x + 1) = f(x)R (it exists because of Prop. n 1A (ii)). Then the conjugacy class of R in GL (k) is called the monodromy of L. n Note that the matrix R itself (unlike the conjugacy class of R, cf. Proposition 1A (ii)) is not well defined since it relies on the choice of the set of solutions f. Definition 2B. Let L be a differential operator of the form L = ∂n+ n−1u ∂j, j=0 j+1 u ∈ C∞(M,C) (M is an elliptic curve). Let f = (f ,...,f ) be a sePt of solutions j 1 n of the equation Lφ = 0 belonging to C∞(Σ,C) whose Wronski matrix is everywhere nondegenerate. Let R ∈ LGL (C) be the matrix such that f(z +τ) = f(z)R(z) (it n exists because of Prop. 1B (ii)). Then the restricted conjugacy class of the element (R,τ) in GL (C) is called the monodromy of L. n The reason for this definition is the following: if g(z) = f(z)Q(z) is another set of solutions (i.e. Q(z) ∈ LGL (C)), then g(z +τ) = g(z)R˜(z), where R˜(z) = n Q−1(z)R(z)Q(z + τ), which corresponds to conjugation of the element (R,τ) ∈ GL (C) by (Q−1,0). Since any set of solutions has the form f(z)Q(z), where Q is n a holomorphic matrix (Proposition 1B, part (ii)), monodromy is well defined, i.e. does not depend on the choice of f. Note that for differential equations on the line there is a canonical choice of a set of solutions f – the set whose Wronski matrix is the identity matrix at a fixed point x of the line (the fundamental system of solutions). The notion of a fundamental 0 system of solutions does not have a natural analogue in two dimensions. Remark. Observe that in Case 2 the monodromy matrix R(z) is always in LGL (C) . Indeed, detR(z) = detW(f)(z+τ), which means that the map z → n 0 detW(f)(z) detW(f)(z+sτ) detR(z) is homotopic to the identity: the homotopy is φ (z) = , s ∈ s detW(f)(z) [0,1]. For a similar reason, in Case 1 if k = R then the determinant of R is always positive. Now we are ready to formulate the main theorem about the local structure of the fibration of L into symplectic leaves. Theorem 3AB. Let L(t),a < t < b be a smooth curve in L. Then L(t) lies inside a single symplectic leaf if and only if the monodromy of L(t) is the same for all t. The proof of this theorem for Case 1 was given in [OK]. Before proving Case 2, let us give a reformulation of the isomonodromic condition in terms of vector bundles and in terms of coadjoint orbits of double loop algebras. Define the rank n vector bundle E on M corresponding to a differential operator L L ∈ L. It will be a flat k-bundle in Case 1 and a holomorphic bundle in Case 2. 8 For every p ∈ M let U be the neighborhood of p such that there exists a set p f = (fp,...,fp) of n solutions of the equation Lφ = 0 defined in U whose Wronski 1 n p matrix is nondegenerate in U . Let the matrices Qpq (belonging to GL (k) in Case p n 1 and LGL (C) in Case 2) be defined by the condition fq = fpQpq. Then Qpq n satisfy the conditions QpqQqp = 1, QpqQqrQrp = 1. Definition 3AB. The vector bundle E is the bundle on M defined by the set of L transition functions Qpq. There is another, more explicit construction of the vector bundle E . Let R be L a monodromy matrix for L. Let Mˆ be the interval [0,1] in Case 1 and the annulus {x+τy ∈ Σ|0 ≤ y ≤ 1} in Case 2. Define the vector bundle E on M as follows. L Take a trivial rank n bundle over Mˆ and glue the two boundaries of Mˆ together: 0 ∼ 1 in Case 1, x ∼ x+τ in Case 2 (this will transform Mˆ into M), identifying the fibers over corresponding points by means of the monodromy matrix R. It is easy to check that the obtained flat (holomorphic) vector bundle over M is isomorphic to E . L Thus, global smooth sections of E can be realized as quasiperiodic vector- L functions on R (respectively on Σ), i.e. as such functions f that f(x+1) = f(x)R (respectively f(z +τ) = f(z)R(z)). LetusnowdefineaffineanddoubleaffineLiealgebras. Letg(M) = C∞(M,gl (k))⊕ n C be the one dimensional central extension of C∞(M,gl (k)) by means of the co- n cycle Ω(f,g) = tr(fDg)ω. In the one-dimensional case it is the usual affine Lie M algebra. In theRtwo-dimensional case it is the double affine algebra considered in [EF]. It is known that the Lie algebra g(M) integrates to a Lie group G(M). ([PS] for Case 1, [EF] for Case 2). The coadjoint representation of this group can be realized as the space of differential operators λD+f (λ ∈ k), where f is a smooth function on M with values in gl (k), in which the action of the group G(M) reduces to the n actionofC∞(M,GL (k))byconjugation(thesocalledgaugeaction): g◦(λD+f) = n λD +Dg ·g−1 +gfg−1. The coadjoint orbit containing the operator ∆ = λD +f will be denoted by O . ∆ The notion of monodromy for operators of the form λD+f (λ 6= 0), where f is matrix-valued, is analogous to that for higher order scalar operators. For D = d/dx this notion is standard; for D = ∂, monodromy is the restricted conjugacy class in GL (C) of an element (g(z),τ) such that there exists a nondegenerate matrix n solution B(z) of the equation λ∂B +fB = 0 defined on the cylinder Σ and such that B(z +τ) = B(z)g(z) [EF]. Consider now the affine linear map ∆ : L → g(M)∗ given by the formula L → D−A , where A is defined by (2.1) (for both Case 1 and Case 2). This map takes L L values in the hyperplane λ = 1. Proposition 4AB. The following three conditions on two differential operators L ,L ∈ L are equivalent: 1 2 (i) L and L have the same monodromy; 1 2 (ii) The flat (respectively holomorphic) vector bundles E and E are isomor- L1 L2 phic. (iii) The points ∆(L ) and ∆(L ) are in the same orbit of coadjoint representa- 1 2 tion of G(M). 9 Proof. It is clear that the monodromy of the operator L is the same as the monodromy of ∆(L). Case 1. The equivalence of (i) and (ii) is obvious; the equivalence of (ii) and (iii) was observed in [F], [RS], [Se]. Case 2. The equivalence of (i) and (ii) is an observation of E.Loojienga (cf. [EF]) (he observed that conjugacy classes in the extended loop group correspond to holomorphic bundles over an elliptic curve). The equivalence of (ii) and (iii) follows from [EF]. (cid:3) Remark. In Case 2 the vector bundle E is always of degree zero since it L is obtained from the trivial bundle on the annulus by gluing with the help of a transition matrix R(z) ∈ LGL (C) which is homotopic to the identity. n 0 Proof of Theorem 3AB for Case 2. The proof given below follows the method of [OK]. Let L(t) be a smooth curve on L. Pick a smooth family of vector-functions ft with a nondegenerate Wronski matrix such that L(t)ft = 0 for all t,i. This i is possible because of Corollary 2AB. Let Rt(z) ∈ LGL (C) be the monodromy n 0 matrix of this set of solutions: it is defined by the formula ft(z +τ) = ft(z)Rt(z). If. We must show that L′(t) is a Hamiltonian vector for any t. Weknow that allelements (Rt(z),τ)areinthe samerestricted conjugacy class in GL (C), i.e. are conjugate to the same element (R(z),τ). Therefore, (Rt(z),τ) is n a smooth curve on the restricted conjugacy class of (R(z),τ). Since the group LGL (C) is the total space of a principal fibration over this restricted conju- n gacy class whose fiber is the centralizer of (R(z),τ) in LGL (C) (this is a finite- n dimensional complex Lie group), the curve (Rt(z),τ) can be lifted to a smooth curve Ct(z) on LGL (C). In other words, there exists a function Ct(z) taking n values in LGL (C) which is smooth in t and satisfies the relation n (2.4) Rt(z) = Ct(z)R(z)(Ct)−1(z +τ). Define a new vector function gt = ftCt. Obviously, its components are still solutions of L(t)φ = 0, and its Wronski matrix is nondegenerate. But now we have an additional property – the monodromy matrix of gt does not depend on t: gt(z +τ) = gt(z)R(z). Let t ∈ (a,b). Let gt = g + (t − t )g′ + O((t − t )2), t → t . Also let 0 0 0 0 L(t) = L + (t − t )L′ + O((t − t )2), t → t . Let us differentiate the relation 0 0 0 L(t)gt = 0 by t at t = t . We get 0 (2.5) Lg′ +L′g = 0. In order to show that L′ is a Hamiltonian vector, we must find a pseudodifferential symbol A such that L′ = V (L) = L(AL) −(LA) L. This is the same as A + + to find an A such that (2.6) Lg′ +(L(AL) −(LA) L)g = 0, + + because the equation Lg′ +Fg = 0 with respect to an n −1-th order differential operator F has a unique solution: F = n c ∂j−1, where c = (c ,...,c ) is equal j=1 j 1 n to F = −(Lg′)W(g)−1 (note that to apPply a differetial operator of order n−1 to a 10 set of n functions h is the same as to multiply the row vector of coefficients of this operator by the Wronski matrix W(h)). Since Lg = 0, equation (2.6) is equivalent to (2.7) L(g′ +(AL) g) = 0. + This means that it is enough to find an A such that (2.8) g′ +(AL) g = 0. + That is, to find an A such that n j−1 (2.9) (AL) = b ∂ , + j Xj=1 where b = (b ,...,b ) is defined as follows: 1 n (2.10) b = −g′W(g)−1. Since g and g′ have the same monodromy matrix, it follows from (2.10) that b is doubly periodic: b ∈ C∞(M,C). i In order to prove the existence of A satisfying (2.9), it suffices to show that the linear map χ : A → L˜ given by χ(A) = (AL) is an isomorphism. But this is + obvious: the coefficients of the operator (AL) , have the triangular form a +P , + i i where P is a differential polynomial in a ,...,a , and hence the coefficients a i 1 i−1 i of the solution of the equation (AL) = Λ, Λ ∈ L˜, can be uniquely determined + recursively starting from a . 1 Only if. Differentiating the equation L(t)ft = 0, we get (2.11) Lf′ +L′f = 0. (we use the shortened notation f for ft). We know that L′ = V (L) for some A. A This implies: (2.12) L(f′ +(AL) f) = 0. + This means that (2.13) f′ +(AL) f = h, + where h satisfies the equation Lh = 0. Let us show that we could have chosen ft in such a way that h = 0. Indeed, let gt be another set of solutions of Lφ = 0 given by (2.14) gt = ft(Ct)−1, where Ct ∈ LGL (C). Substituting (2.14) in (2.13), we get n (2.15) g′C +gC′ +(AL) gC = h +